This paper demonstrates the importance of three-dimensional(3-D)piezoelectric coupling in the electromechanical behavior of piezoelectric devices using three-dimensional finite element analyses based on weak and stron...This paper demonstrates the importance of three-dimensional(3-D)piezoelectric coupling in the electromechanical behavior of piezoelectric devices using three-dimensional finite element analyses based on weak and strong coupling models for a thin cantilevered piezoelectric bimorph actuator.It is found that there is a significant difference between the strong and weak coupling solutions given by coupling direct and inverse piezoelectric effects(i.e.,piezoelectric coupling effect).In addition,there is significant longitudinal bending caused by the constraint of the inverse piezoelectric effect in the width direction at the fixed end(i.e.,3-D effect).Hence,modeling of these effects or 3-D piezoelectric coupling modeling is an electromechanical basis for the piezoelectric devices,which contributes to the accurate prediction of their behavior.展开更多
1 Introduction and preliminaries The aim of this note is to prove the following basic theorem: Let (Ω, σ, u) be a probability space, (B, ‖·‖) a weakly compactly generated Banach space, and a mapping V from Ω...1 Introduction and preliminaries The aim of this note is to prove the following basic theorem: Let (Ω, σ, u) be a probability space, (B, ‖·‖) a weakly compactly generated Banach space, and a mapping V from Ω to B be a weak random element, then there exists a unique strongly measurable random element V from Ω to B under the sense of almost sure equality such that (?) is weakly equivalent to the weak random dement V. This theorem itself not only removes the limitation that the weak random element considered in a theorem due to Lewis is bounded, but also has many applications to probability theory in Banach spacest. As an example of applications, we give a theorem of properties of the reproducing kernel Hilbert spaces for weak twofold weak random elements.展开更多
For a double array of independent random elements {Vmn,m ≥ 1,n ≥ 1} in a real separable Banach space,conditions are provided under which the weak and strong laws of large numbers for the double sums mi=1 nj=1Vij,m ...For a double array of independent random elements {Vmn,m ≥ 1,n ≥ 1} in a real separable Banach space,conditions are provided under which the weak and strong laws of large numbers for the double sums mi=1 nj=1Vij,m ≥ 1,n ≥ 1 are equivalent.Both the identically distributed and the nonidentically distributed cases are treated.In the main theorems,no assumptions are made concerning the geometry of the underlying Banach space.These theorems are applied to obtain Kolmogorov,Brunk–Chung,and Marcinkiewicz–Zygmund type strong laws of large numbers for double sums in Rademacher type p（1 ≤ p ≤ 2） Banach spaces.展开更多
基金supported by the Japan Society for the Promotion of Science under KAKENHI Grant Nos.19F19379 and 20H04199。
文摘This paper demonstrates the importance of three-dimensional(3-D)piezoelectric coupling in the electromechanical behavior of piezoelectric devices using three-dimensional finite element analyses based on weak and strong coupling models for a thin cantilevered piezoelectric bimorph actuator.It is found that there is a significant difference between the strong and weak coupling solutions given by coupling direct and inverse piezoelectric effects(i.e.,piezoelectric coupling effect).In addition,there is significant longitudinal bending caused by the constraint of the inverse piezoelectric effect in the width direction at the fixed end(i.e.,3-D effect).Hence,modeling of these effects or 3-D piezoelectric coupling modeling is an electromechanical basis for the piezoelectric devices,which contributes to the accurate prediction of their behavior.
文摘1 Introduction and preliminaries The aim of this note is to prove the following basic theorem: Let (Ω, σ, u) be a probability space, (B, ‖·‖) a weakly compactly generated Banach space, and a mapping V from Ω to B be a weak random element, then there exists a unique strongly measurable random element V from Ω to B under the sense of almost sure equality such that (?) is weakly equivalent to the weak random dement V. This theorem itself not only removes the limitation that the weak random element considered in a theorem due to Lewis is bounded, but also has many applications to probability theory in Banach spacest. As an example of applications, we give a theorem of properties of the reproducing kernel Hilbert spaces for weak twofold weak random elements.
基金supported by the Vietnam Institute for Advanced Study in Mathematics(VIASM)the Vietnam National Foundation for Sciences and Technology Development NAFOSTED(Grant No.101.01.2012.13)supported by NAFOSTED(Grant No.101.03.2012.17)
文摘For a double array of independent random elements {Vmn,m ≥ 1,n ≥ 1} in a real separable Banach space,conditions are provided under which the weak and strong laws of large numbers for the double sums mi=1 nj=1Vij,m ≥ 1,n ≥ 1 are equivalent.Both the identically distributed and the nonidentically distributed cases are treated.In the main theorems,no assumptions are made concerning the geometry of the underlying Banach space.These theorems are applied to obtain Kolmogorov,Brunk–Chung,and Marcinkiewicz–Zygmund type strong laws of large numbers for double sums in Rademacher type p（1 ≤ p ≤ 2） Banach spaces.
